Questions tagged [sheaf-cohomology]

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5
votes
1answer
165 views

Elementary proof of the exactness of Čech complex associated to a hypercovering (“Illusie's Conjecture”)

Let $\mathcal{E}$ be a sheaf of abelian groups on a topological space (or a site). For an open covering $\mathfrak{U} = (U_i)_i$, it is well known that the augmented Čech complex $0 \to \mathcal{E} \...
3
votes
0answers
61 views

Sheaf cohomology of Grassmannian G(2,4) with values in twisted tautological bundles over an arbitrary field

Let k be an arbitrary field. Let $G(2,4)_k$ be the Grassmannian of 2-planes in 4-space over that field. Let $\mathcal{E}$ be the tautological quotient bundle on the Grassmannian. I am trying to ...
12
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0answers
414 views

Is there a concrete application of topos theory?

The notion of topos was originally formulated in SGA 4 in the context of attacking the Weil conjectures. This formalism turned out to be unnecessary for the purposes of proving those conjectures. But ...
2
votes
1answer
191 views

Help about “Varieties with small Dual Varieties” by L.Ein

I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...
5
votes
1answer
129 views

Cohomology of doubly pinched torus via spectral sequences

Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
3
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0answers
184 views

Spectral sequence from resolution of condensed abelian groups

I am watching Scholze's and Clausen's masterclass on Condensed Mathematics and I don't understand or can find any references on something they said. You have a resolution $$ \dots \to \mathbb{Z}[\...
1
vote
1answer
229 views

Relationship between $H^1(X, \mathbb{T})$ and complex line bundles

Let $X$ be a compact metric space and consider the sheaf cohomology group $H^1(X, \mathbb{T})$. From a class in $H^1(X, \mathbb{T})$, I can get a principal $\mathbb{T}$-bundle over $X$ and from this, ...
3
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0answers
166 views

Sheaf cohomology on the formal completion of $\mathbb{P}^1_k$ at its north pole

Let $k$ be a field and let $\mathbb{P}^1_k$ be the projective line over $k$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbb{P}^1_k$. The curve $\mathbb{P}^1_k$ is recovered by two affine subschemes ...
0
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0answers
137 views

Dimension of global holomorphic sections of a line bundle

Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...
3
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0answers
99 views

Elementary reference for Borel-Moore/locally finite homology

There is a homology theory called "Borel-Moore" or "locally finite" homology, which can either be constructed by using locally-finite chains or by more advanced sheaf-theoretic ...
1
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0answers
57 views

$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality

This is a question arising when I am reading M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. ...
1
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0answers
107 views

Sections of the structure sheaf of a partial flag variety on big cell

Let $G$ be a connected split reductive group over a (non-archimedean) field $K$ of char 0 with split maximal torus $T$ and a standard parabolic $P$. Denote by $W$ the Weyl group of $G$ and by $W_P$ ...
4
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0answers
124 views

Continuity property for Čech cohomology

Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...
0
votes
0answers
43 views

Image of first group of Cech cohomology in the second group of De Rham cohomology

Let $M$ be a smooth manifold and $\mathcal{U}$ an open good cover of $M$. If $\underline{\mathbb{Z}}$ denotes sheaf of locally constant functions, $C^{\infty}(U) := C^{\infty}(U, \mathbb{R})$ and $\...
2
votes
0answers
104 views

Are flasque sheaves exactly the retracts of “canonically” flasque sheaves?

Let $X$ be a topological space. Let $X^\delta$ denote the space whose elements are the points of $X$, and which is equipped with the discrete topology. There is a continuous map $i : X^\delta\to X$ ...
1
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0answers
133 views

Explicit map between $\check{H}^1(M,\underline{\mathbb{R}})$ and $H^1(M,\mathbb{R})$

Is there a way to construct an explicit isomorphism between Cech cohomology and singular cohomology on a smooth manifold for degree 1? If yes can this be extended to higher degee?
0
votes
1answer
93 views

Connecting homomorphism in Cech cohomology

Let $M$ be a smooth manifold and $\mathcal{U}$ be a good open cover of $M$. If I have an exact sequence of sheaves $$0 \longrightarrow A \stackrel{f}\longrightarrow B \stackrel{g}\longrightarrow C \...
4
votes
0answers
175 views

Does cohomology and base change hold if supported at a point?

I have a flat, quasicompact, and separated map $p : X \to \mathbb{A}^1$ and I know that $R^i p_* \mathcal{O}_X$ vanishes everywhere except possibly $0 \in \mathbb{A}^1$. Q1: Does "cohomology and ...
3
votes
1answer
265 views

Elementary way to compute Hodge numbers of Grassmanian

I know that by using Hodge decomposition and the fact that Schubert cells are Hodge cycles you can compute the Hodge numbers of Grassmanian but is there a more elementary way to compute sheaf ...
8
votes
0answers
390 views

intuition about perverse sheaves

firstly, I would know if my very basic intuition on perverse sheaves is correct . secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves . my intuition ...
1
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0answers
90 views

Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...
4
votes
0answers
70 views

Serre vanishing on one-point blow-ups

This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry. Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...
1
vote
1answer
107 views

Relation between characteristic cycle and singular support of constructible sheaf

Let $M$ be a real analytic manifold. Let $F$ be an object of the bounded derived category of sheaves on $M$ with real constructible cohomology sheaves. Let $CC(F)$ denote the characteristic cycle of $...
1
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0answers
63 views

Compute Cech cohomology with two open sets

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ be a sheaf of $\...
4
votes
1answer
202 views

Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$

I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3) and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON $\...
1
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0answers
140 views

Cohomology of a family of twisted cubic curves (Hartshorne III, 12.9.2)

I'm trying to understand following Example in Hartshorne (Chapter III, Example 9.8.2 & Example 12.9.2): Let $X_1 \subset \mathbb{P}^{3}$ be a twisted cubic curve not containing the point $(0:...:1)...
4
votes
2answers
292 views

Sheaf cohomology commutes with colimits of sheaves

Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F_{\alpha})_{\alpha \in I}$ a direct system of $O_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof ...
5
votes
0answers
245 views

Calculation in prismatic cohomology

In the standard references for prismatic cohomology, most theorems are proved in a local context (i.e. with completeness assumptions), and the devissage to the global case (i.e. smooth proper ...
12
votes
1answer
283 views

Sheaves in combinatorics and discrete geometry

I am looking for examples for the application of sheaves, sheaf-like constructions or the (co)homology of sheaves to problems in combinatorics and discrete geometry. For example given a poset $(P,\...
1
vote
0answers
111 views

Surjectivity of multiplicative map (in more specific case)

(I have asked the question Surjectivity of multiplicative map. I ask here the more specific case.) Let $S$ be a smooth complex algebraic surface, and $D$ be a divisor on $S$ such that $D^2>0$ and $...
0
votes
0answers
124 views

Surjectivity of multiplicative map

Let $S$ be a smooth complex algebraic surface, and $\mathcal{F}$ be a coherent sheaf on $S$. I want to consider $W = S \times S$ and the coherent sheaf $\mathcal{G} = \mathcal{F} \boxtimes \mathcal{F}...
1
vote
1answer
77 views

Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$?

Let $X$ be a locally Noetherian scheme and $i:Z\to X$ be an immersion of closed subschemes. Let $\mathcal{F},\mathcal{G}$ be two etale abelian sheaves over $X_{et}$. We can define the subsheaf $\...
4
votes
0answers
101 views

Understanding a step in proof of sheaf version Verdier duality

Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss. So all proofs I can find factors through a particular statement, which goes ...
3
votes
1answer
98 views

Subspace inclusion with non-vanishing higher direct images

I'm looking for concrete topological intuition for the derived pushforward. Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...
1
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0answers
42 views

local acyclicity when restricting to an hypersurface

Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$. Suppose that $K$ is locally ...
2
votes
0answers
109 views

A infinity structure on Yoneda Ext group

I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...
1
vote
1answer
176 views

Injectivity of the cohomology map associated to the pullback of line bundles

Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just ...
1
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0answers
265 views

Čech-Alexander complex in computing (crystalline/prismatic) cohomology

I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology). They seemed to be introduced as a method ...
3
votes
0answers
141 views

Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology

There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions ...
1
vote
1answer
346 views

Pullback map on global sections surjective

Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism! Let $\mathcal{L}$ ...
1
vote
0answers
114 views

Group cohomology of sheaves under closed immersion

Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...
4
votes
1answer
351 views

The Yoneda pairing, hypercohomology, and cup product

Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
2
votes
1answer
311 views

Very weak Riemann-Roch on curves (by J. Kollar)

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14): 1.13 (Very weak Riemann-Roch on curves)...
3
votes
1answer
285 views

Does local cohomology commute with pullback?

Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open. For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\...
8
votes
0answers
217 views

Global functions on a product of schemes over artinian ring

For a morphism of schemes $f:X\to S$ with $S=\text{Spec}(R)$ affine, let's write $A(X)=H^0(X,\mathcal{O}_X)$. I'm interested in the morphism of $R$-algebras $$ c:A(X)\otimes_R A(Y)\to A(X\times_SY) $$ ...
3
votes
0answers
144 views

Sheaf cohomology of the complement of a schubert variety

Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...
10
votes
3answers
933 views

How to compute the cohomology of a local system?

Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well. Suppose that we are ...
2
votes
1answer
207 views

Pushforward in Compactly Supported Cohomology

Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
8
votes
1answer
287 views

Does the sheaf $\mathcal{O}^*$ on a complex manifold have an acyclic cover?

Let $X$ be a complex manifold and let $\mathcal{O}^*$ be the sheaf nonvanishing holomorphic functions on it. Does it have an acyclic cover? That is, a cover for which all open sets and all ...
1
vote
1answer
233 views

Multiplicative structure for sheaf cohomology of flag varieties

Let $F,F'$ be two locally free sheaves on a smooth complex algebraic variety. There is a cup-product $H^i(X, F) \otimes H^j(X,F') \to H^{i+j}(X,F \otimes F')$. In particular if $F$ is the sheaf of ...

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